Is Zero a Factor of Zero?

Generally speaking, if \(a \times b = c\), then \(a\) and \(b\) are factors of \(c\). This concept appears at the secondary level in two contexts: The factors of positive integers, and the factors of a polynomial. If we limit the domain and range to be positive integers, for instance, the factors of 7 are {1, 7} while the factors of 8 are {1, 2, 4, 8}.

The term “prime number” is often defined in these terms: A prime number has only two distinct factors, itself and 1. This does raise the issue of whether 1 is itself prime, which was an open question for a long time. Eventually, mathematicians decided that positive integers should be broken into three groups, not two. 1 has one factor, primes have two factors, and composites have three or more factors.

We can select the domain and range differently, though. Still excluding zero, if the domain and range are all integers, then the factors of 7 are {1, 7, -1, -7}, while the factors of 8 are {1, 2, 4, 8, -1, -2, -4, -8}. This is a trivial expansion, although it changes the set of perfect numbers. A perfect number is a number which is the sum of its proper factors (or, said another way, a perfect number is half the sum of its factors). The smallest perfect number by the standard domain and range is 6: {1, 2, 3, 6}. With this modification, no numbers are perfect, since the sum of the factors will always be zero.

Again excluding zero, if the doman and range are all real numbers, then the factors of all real numbers are all real numbers. To see this, start with the factors of 1. \([\forall a \in \mathbb{R}, a \times 1/a = 1]\). For any real number, multiply it by its reciprocal to get 1. To adjust this for any other real number, replace 1 by that number. That is, \([\forall a, n \in \mathbb{R}, a \times n/a = n]\).

So what about zero? Let’s limit our doman and range to all integers. Two observations can be made immediately:

Zero is not a factor of any non-zero number because \(n/0\) is undefined for any number other than zero.

All integers other than zero are factors of zero because \(0n = 0\) for all numbers.

But is zero a factor of zero?

Wolfram|Alpha defines “factor” thus: “A factor is a portion of a quantity, usually an integer or polynomial that, when multiplied by other factors, gives the entire quantity. The determination of factors is called factorization (or sometimes ‘factoring’). In number theoretic usage, a factor of a number n is equivalent to a divisor of n. The divisors of a number n are given in the Wolfram Language by the command Divisors[n]. In elementary education, the term “factor” is sometimes used to mean proper divisor, i.e., a factor of n other than the number n itself. However, as a result of the confusion this practice creates and its inconsistency with the mathematical literature, it should be discouraged.”

By the first portion of this, zero is a factor of zero because \(0n = 0\) for all numbers. However, is zero a divisor of zero? \(0/0\) is indeterminate, which is crucially different from being undefined. \(1/0\) is undefined because there is no value such that \(0n = 1\); \(0/0\) is indeterminate precisely because \(0n = 0\) for all \(n\), so we don’t know which \(n\) led to \(0/0\).

Using Wolfram’s Divisors function isn’t decisive because it only returns positive integers regardless. Divisors[-1] returns {1}, for instance. Divisors[0] returns “all non-zero integers”, which is a little sloppy (“all positive integers” would be just as good an answer without confusing the matter with negative integers; after all, -1 is clearly a factor of -1, if negative integers are to be included).

To me, the specific answer to the question (if there is one) is less important than the ramifications of playing with the domain and range of the concept of factors. The important thing here is that we need to make sure our definitions are rigorous and account for contradictions.

In this case, I would say that the first part of Wolfram’s definition is conceptually appropriate, and is where I started: If \(a \times b = c\) then \(a\) and \(b\) are factors of \(c\). By this definition, zero is unambiguously a factor of zero.